Gear transmission systems are used for transmitting motion by means of successively engaging teeth. During operation of a machine, one flank of a tooth of one gear meshes with one flank of a tooth of another gear. In order to obtain optimal meshing between the teeth of the gears, the tooth profile of the teeth of the gears is often optimized. In the past, tooth profile optimization has already extensively been studied in order to optimize tooth contact between meshing gears, which has led to different known tooth profile optimization methods.
For example, in SI 21810 Hlebanja et al. describe an internal gearing pair with S-gearing which comprises a gearwheel with internal teeth 1 and a gearwheel with external teeth 2 featuring lateral tooth profiles which conforms to a curved semi-symmetrical S-shaped fitting track or contact line 5 (see FIG. 1). The geometrical shape of the fitting track or contact line 5 is determined by the slope in its center αC=20°+/−2° and increasing curve factor between the center and the ends. Typical and predefined are the point A at the beginning of fitting, the point E at the end of fitting and the center of the fitting curve, which coincides with the kinematic pole C of the gearwheel pair. The distance between the starting point of fitting A and the final point of fitting E on one side, as well as the center 9 of the worm wheel gearing on the other corresponds to the module value. The distance of points A and E from the vertical axis of symmetry 10 of the gearwheels depends on the angle αE=38°+/−1°, which results between the straight line E-C, or on the other side the straight line A-C with the axis of symmetry of the wormwheel gearing 9.
Another example is described in EP 0 974 016, which discloses a gearing system which includes a pair of gears (see FIG. 2). The tooth profile of the first gear 11 has three portions: a concave portion 11b lying within the dedendum of the first gear 11, a convex portion 11c lying within the addendum of the first gear 10 and a transition zone 11a disposed between the concave portion 11b and convex portion 11c. Similarly, the tooth profile of the mating gear 12 has three portions; a concave portion 12b lying within the dedendum of the mating gear 12, conjugate to the convex portion 11c of the tooth profile of the first gear 11; a convex portion 12c lying within the addendum of the mating gear 12, conjugate to the concave portion 11b of the tooth profile of the first gear 11; and a transition zone 12a disposed between the concave portion 12b and convex portion 12c. The pair of gears 11, 12 may be designed such that no contact between meshing teeth is made along the transition zones 11a, 12a. 
A drawback of the above described methods is that they only allow optimizing one tooth contact, i.e. one flank of a tooth. Therefore, these methods may be less suitable to be used for idler gears, i.e. for gears which are interacting with two other gears. The teeth of idler gears on one side, also referred to as first flank, make contact with teeth of a first other gear and on the other side, also referred to as second flank, make contact with teeth of a second other gear. The first tooth flanks of the teeth of the idler gear may be subject to a different load distribution than the second tooth flanks of the teeth. For example, in case of a planetary gear unit comprising planet gears for mutual interaction with a sun gear and a ring gear, the tooth flanks of the planet gear teeth at the side of the sun gear are subject to a different load distribution than the tooth flanks of the planet gear teeth at the side of the ring gear.
Hence, when known tooth profile optimization methods as described above are performed on idler gears, they would only allow to optimize contact between a first flank of the teeth of the idler gear with teeth of another gear, and may thus still have drawbacks for the other tooth contact.
Disadvantages hereof may, for example, be that this may limit the allowable torque of the gear transmission systems, for example because of low tooth root strength. This may furthermore have disadvantages with respect to the dynamic behavior of the gear transmission systems.
In both the above described examples, the tooth profiles have a convex/concave geometry. Also examples are known of asymmetric tooth profiles. One example of such asymmetric tooth profiles and how to design these tooth profiles is described by Alexander Kapelevich in “Geometry and design of involute spur gears with asymmetric teeth. The method described in this paper is based on the following considerations. A first one is that for an external gearing a larger pressure angle is proposed for the drive tooth side but not for the coast tooth side. A second consideration is that the parameters of asymmetric gears are defined independently form any generating rack parameters. On the contrary, the choice of the generating rack parameters is based on results of the asymmetric gear synthesis. The generating rack parameters for the pinion and the gear are optimized independently and, as a rule, they are different.
Again, similar as the methods described above, this method only allows optimizing one tooth contact, i.e. one flank of a tooth, and may thus be less suitable for being used for idler gears.